Average Error: 0.6 → 0.6
Time: 3.4s
Precision: 64
\[\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\]
↓
\[\sqrt{re \cdot re + im \cdot im}\]
\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}
↓
\sqrt{re \cdot re + im \cdot im}
double f(double re, double im) {
        double r1142212 = re;
        double r1142213 = r1142212 * r1142212;
        double r1142214 = im;
        double r1142215 = r1142214 * r1142214;
        double r1142216 = r1142213 + r1142215;
        double r1142217 = sqrt(r1142216);
        return r1142217;
}
↓
double f(double re, double im) {
        double r1142218 = re;
        double r1142219 = r1142218 * r1142218;
        double r1142220 = im;
        double r1142221 = r1142220 * r1142220;
        double r1142222 = r1142219 + r1142221;
        double r1142223 = sqrt(r1142222);
        return r1142223;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.6

    \[\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\]

  2. Final simplification0.6

    \[\leadsto \sqrt{re \cdot re + im \cdot im}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))